# Some New Hilbert's Type Inequalities

- Chang-Jian Zhao
^{1}Email author and - Wing-Sum Cheung
^{2}

**2009**:851360

https://doi.org/10.1155/2009/851360

© C.-J. Zhao and W.-S. Cheung. 2009

**Received: **25 December 2008

**Accepted: **24 April 2009

**Published: **5 May 2009

## Abstract

Some new inequalities similar to Hilbert's type inequality involving series of nonnegative terms are established.

## 1. Introduction

In recent years, several authors [1–10] have given considerable attention to Hilbert's type inequalities and their various generalizations. In particular, in [1], Pachpatte proved somenew inequalities similar to Hilbert's inequality [11, page 226] involving series of nonnegative terms. The main purpose of this paper is to establish their general forms.

## 2. Main Results

In [1], Pachpatte established the following inequality involving series of nonnegative terms.

Theorem 2 A.

We first establish the following general form of inequality (2.1).

Theorem 2.1.

Proof.

This completes the proof.

Remark 2.2.

Taking and changing , , and into , , and respectively, and with suitable changes, (2.11) reduces to Pachpatte [1, inequality (1)].

In [1], Pachpatte also established the following inequality involving series of nonnegative terms.

Theorem 2 B.

Inequality (2.12) can also be generalized to the following general form.

Theorem 2.3.

Proof.

The proof is complete.

Remark 2.4.

Taking and changing , , and into , , and respectively, and with suitable changes, (2.21) reduces to Pachpatte [1, Inequality ( 7)].

Theorem 2.5.

Proof.

Proceeding now much as in the proof of Theorems 2.1 and 2.3, and with suitable modifications, it is not hard to arrive at the desired inequality. The details are omitted here.

Remark 2.6.

In the special case where and , Theorem 2.5 reduces to the following result.

Theorem 2 C.

This is the new inequality of Pachpatte in [1,Theorem 4].

Remark 2.7.

Taking and in Theorem 2.5, and in view of , we obtain the following theorem.

Theorem 2 D.

This is the new inequality of Pachpatte in [1,Theorem 3].

## Declarations

### Acknowledgments

Research is supported by Zhejiang Provincial Natural Science Foundation of China(Y605065), Foundation of the Education Department of Zhejiang Province of China (20050392). Research is partially supported by the Research Grants Council of the Hong Kong SAR, China (Project no. HKU7016/07P) and a HKU Seed Grant forBasic Research.

## Authors’ Affiliations

## References

- Pachpatte BG:
**On some new inequalities similar to Hilbert's inequality.***Journal of Mathematical Analysis and Applications*1998,**226**(1):166–179. 10.1006/jmaa.1998.6043MATHMathSciNetView ArticleGoogle Scholar - Handley GD, Koliha JJ, Pečarić JE:
**New Hilbert-Pachpatte type integral inequalities.***Journal of Mathematical Analysis and Applications*2001,**257**(1):238–250. 10.1006/jmaa.2000.7350MATHMathSciNetView ArticleGoogle Scholar - Gao M, Yang B:
**On the extended Hilbert's inequality.***Proceedings of the American Mathematical Society*1998,**126**(3):751–759. 10.1090/S0002-9939-98-04444-XMATHMathSciNetView ArticleGoogle Scholar - Jichang K:
**On new extensions of Hilbert's integral inequality.***Journal of Mathematical Analysis and Applications*1999,**235**(2):608–614. 10.1006/jmaa.1999.6373MATHMathSciNetView ArticleGoogle Scholar - Yang B:
**On new generalizations of Hilbert's inequality.***Journal of Mathematical Analysis and Applications*2000,**248**(1):29–40. 10.1006/jmaa.2000.6860MATHMathSciNetView ArticleGoogle Scholar - Zhong W, Yang B:
**On a multiple Hilbert-type integral inequality with the symmetric kernel.***Journal of Inequalities and Applications*2007,**2007:**-17.Google Scholar - Zhao C-J:
**Inverses of disperse and continuous Pachpatte's inequalities.***Acta Mathematica Sinica*2003,**46**(6):1111–1116.MATHMathSciNetGoogle Scholar - Zhao C-J:
**Generalization on two new Hilbert type inequalities.***Journal of Mathematics*2000,**20**(4):413–416.MATHMathSciNetGoogle Scholar - Zhao C-J, Debnath L:
**Some new inverse type Hilbert integral inequalities.***Journal of Mathematical Analysis and Applications*2001,**262**(1):411–418. 10.1006/jmaa.2001.7595MATHMathSciNetView ArticleGoogle Scholar - Handley GD, Koliha JJ, Pečarić J:
**A Hilbert type inequality.***Tamkang Journal of Mathematics*2000,**31**(4):311–315.MATHMathSciNetGoogle Scholar - Hardy GH, Littlewood JE, Pólya G:
*Inequalities*. Cambridge University Press, Cambridge, UK; 1934.Google Scholar - Németh J:
**Generalizations of the Hardy-Littlewood inequality.***Acta Scientiarum Mathematicarum*1971,**32:**295–299.MATHGoogle Scholar - Beckenbach EF, Bellman R:
*Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Band 30*. Springer, Berlin, Germany; 1961:xii+198.Google Scholar

## Copyright

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